Singularity in signal theory

Abstract

A discrete-time random signal is singular if its values are singular random variables defined by a distribution function continuous but with a derivative equal to zero almost everywhere. Singular random signals can be obtained at the output of some linear filters when the input is a discrete-valued white noise. Sufficient conditions for singularity are established. In particular it is shown that if the poles of the filter are inside a circle called the circle of singularity and if the input is white and discrete-valued the output is singular. Computer experiments using histograms at different scales exhibit the structure of singular signals. The influence of input correlation is also analysed. It is shown that when the input is not white, but has a specific Markovian structure, the output can be singular. This is also verified by computer experiments. Finally, mixtures of singular and discrete-valued random signals are analysed.